The most important example of madness, or loss of sanity, in The Hunger Games is Katniss's mother. The residents of District 12 live in perpetual poverty, starving and mining for coal. There are few safety measures taken, since human life is so valueless, and during Katniss's youth, her father dies in a mine explosion. After that, her mother fell into a a catatonic state, unable to work or feed her children. Katniss was forced to fend for herself and her sister Prim alone.

She didn't do anything but sit propped up in a chair or, more often, huddled under the blankets on her bed, eyes fixed on some point in the distance.(Collins, The Hunger Games, Google Books)

Her mother's temporary insanity forced Katniss to grow up fast, and become independent from the typical working world of District 12. With her sister's life depending on her, Katniss formed friendships, learned to hunt, and became mentally strong enough to live through the Games. When Katniss leaves, she tells her mother in strong terms that she cannot afford to lose her mind again, since Prim now only has her mother to depend on.

As belarafon explained before, Katniss mother is the best example from the book to explain madness and insanity. But I think that besides her, there are lots of examples shown during the games themselves.

The young children that play ("PLAY" or better described FIGHT) the games, are the ones, that, in my opinion, show the most features of a mad person.

They are forced to kill, to fight for their survival.

Cato for example, in the end, gets so crazy from killing the others, and the fear of being killed himself, that he doesn't care how, but he is determined to kill Katniss. I imagine his look like eyes in fire, passion for destruction.